Dark Matter

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Visible stars are not the only matter in space; in fact, the majority of matter in the universe is known as dark matter due to the fact that it cannot be directly observed. Instead, astronomers and cosmologists infer the existence of dark matter due to its gravitational influence on visible matter. Explaining the physical nature and spatial distribution of dark matter in the universe is one of the main open issues in contemporary physics.

In general, the matter in the universe can be classified into two categories: luminous matter, which emits and absorbs light; and dark matter, which neither absorbs nor emits light. Luminous matter can be stars, glowing gas in nebulae, or even planets, but regardless of the form, all known luminous matter is composed of the particles of the Standard Model: quarks, electrons, photons, and so on. In contrast, it is not known what kind of particles comprise dark matter. Some theories predict that dark matter is in fact composed of ordinary forms of matter that is for some reason undetectable, while others account for the strange gravitational signatures of dark matter via hypothetical particles such as the supersymmetric particles of string theory or the axion. The leading cosmological theory, called the \(\Lambda\text{CDM}\) model, predicts dark matter in the form of some undiscovered weakly interacting massive particle or WIMP for short.

Mapping the amount and distribution of luminous matter in the universe does not pose a technical challenge: after all, it is easy to visually record the number and location of stars in the sky, for example. It is much harder to figure out the amount and distribution of dark matter. Some telescopes and satellites map the distribution of dark matter by examining how it affects the light we see from luminous matter. Other telescopes search for signals as to what exactly dark matter might be made of. Particle accelerators and other experiments also attempt to create and measure dark matter on earth.

Contents

The Distribution of Visible Matter in Galaxies

Investigation into the nature of dark matter began with the observation of visible matter, since dark matter is identified by deviations from the gravitational effects accounted for by visible matter; specifically, observations of stars in galaxies in the context of Newtonian gravitation. Since stars in spiral galaxies like the Milky Way orbit around the center of the galaxy they belong to, the center of a galaxy must have a mass much greater than the mass of a single star. Spiral galaxies do in fact have a large bulge of mass in their centers around which the stars in a galaxy orbit. Outside the bulge is the galactic disk composed of spiral arms and finally a halo surrounding the galaxy.

Image of the spiral galaxy NGC1187 taken from [1], demonstrating the structure described above.

Knowledge of the distribution of visible matter in the outer parts of the disk and halo turns out to be essential to predicting the existence of dark matter. Since visible matter is luminous, the distribution of visible matter can be inferred from the surface brightness curve of a galaxy, which gives the intensity of starlight as a function of radius. Outside the galactic bulge, the surface brightness \(I(r)\) follows an exponential curve:

\[I(r) = I_0 e^{-r/r_0}.\]

where \(I_0\) is a constant with units of intensity that depends on the galaxy, and \(r_0\) is some distance that is approximately \(104\) light years. A light year is equal to the distance that light travels in one year, which is equal to 5.88 trillion miles or \(9.4 \times 10^{15} \text{ m}\).

Since the surface brightness tells us how many stars there are, the surface density \(D(r)\) of matter at any particular radius from the center also has a fall-off given by the same form of equation:

\[D(r) = D_0 e^{-r/r_0},\]

where again \(D_0\) is some constant now with units of density that depends on the galaxy.

Astronomers can observe this exponential decrease in surface density in most galaxies at least out to 4 or 5 times \(r_0\). Measurements of the surface density of luminous matter allow physicists to measure the amount and distribution of visible matter in a galaxy.

The surface density of luminous matter in a galaxy has units of mass per area and obeys the equation:

\[D(r) = D_0 e^{-r/r_0}\]

where \(D_0\) is some constant with units of mass per area. Suppose we have some galaxy where \(D_0 = .25 \text{ kg}/\text{m}\). What is \(D(2r_0)\) in units of \( \text{ kg}/\text{m}\), to the nearest thousandth?

Predicting the Behavior of Visible Matter in Galaxies

The formula for surface density of luminous matter in a galaxy implies that if we look at the stars towards the edges of a galaxy, the amount of mass inside the orbits of these stars is almost the entire mass of the galaxy. This is because the exponential fall-off of the surface density, which means only exponentially few stars are outside large radii. Therefore, well outside the central bulge, the gravitational mass attracting stars in the outer rim of the galactic disk and galactic halo is essentially a constant as we move out in radius from the galactic center (assuming all the matter is luminous).

Even though the radius of a typical star’s orbit around a galactic bulge is vastly larger than the distances in the solar system, Newton’s laws of gravity still apply to the orbits of stars. The magnitude of the Newtonian gravitational force \(F_g\) on a star of mass \(m\) well outside the galactic bulge is given as a function of radius \(r\) from the center of the galaxy by:

\[F_g = G\frac{mM}{r^2}\]

where \(G\approx 6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2\) is Newton’s gravitational constant and \(M\) is the total mass of the galaxy.

Assuming these stars move in circular orbits around the center, they are bound by a centripetal force given by:

\[F_c = \frac{mv^2}{r}.\]

Since the gravitational force is the only force keeping the stars in orbit, it is equal to the centripetal force. Equating the two yields a formula for the velocity of these far-away stars as a function of radius \(r\) from the center of the galaxy:

\[v(r) = \sqrt{\frac{GM}{r}}.\]

The expected velocity of stars in a galaxy should therefore decrease towards the outer edges of galaxies.

It is possible for astronomers to measure the velocity of stars in the outer parts of the galaxy as they orbit and compare them to the above prediction. This is traditionally done using Doppler shift methods. Deviations from the predicted velocity profile are an example of astronomical evidence in support of the existence of dark matter.

How is the orbital velocity of stars at the outer edges of galaxies measured observationally using the Doppler effect?

Answer:

Consider the sun, which appears to emit primarily yellow light during the daytime. In reality, the sun emits many different colors of light. Using a prism, one can split apparently yellow sunlight into a rainbow of colors. Intriguingly, the spectrum of each color present in sunlight reveals some dark gaps at some frequencies, as shown in the figure below:

The intensity of each frequency in the visible spectrum present in sunlight reveals dark gaps called Fraunhofer lines. Image taken from [2].

The dark lines occur because the outer layers of gas in the sun actually absorb certain colors of light, preventing them from reaching astronomical instruments and human eyes. These spectral lines are characteristic of all stars, and their location in the visible spectrum can be predicted accurately by which gases are present in a star.

By measuring the spectrum of a star and comparing the absorption lines to where they should be, the velocities of stars can be measured. A star moving towards an observer will have the dark lines shifted towards the blue end of the spectrum and vice versa, as demonstrated by the below figure:

Redshifting and blueshifting of spectral lines of stars [3].

A formula in astronomy relates the magnitude of the redshift or blueshift to the magnitude and direction of the velocity of the relevant star.

Signs of Dark Matter in the Universe

Using the Doppler shift technique, the observed velocities of stars orbiting galaxies and predicted velocities from Newtonian gravitation can be compared. These predicted velocities are based off of accounting for all of the luminous mass in a galaxy, via measuring the surface density of luminous matter. The following is a graph of the observed rotational velocities of stars far from the galactic bulge as a function of radius for a typical spiral galaxy, often called a galaxy rotation curve. The radial distance is in kiloparsecs, where 1 parsec is 3.26 light years:

This plot demonstrates that the error bars on the observed velocities are reasonably small compared to the velocities. Below, the observed velocities are compared to the Newtonian prediction \(v = \sqrt{\frac{GM}{r}}\), where curve A is the Newtonian prediction and curve B is the experimental data:

The Newtonian prediction for orbital velocity is a poor model for the experimental data [5].

The peak at the beginning of the curves above are due to the large numbers of stars in the bulge. Near the central bulge there are so many stars that the gravitational mass attracting a star at radius r away from the galactic center changes significantly as the radius changes.

Outside the bulge, however, the observed versus predicted curves look nothing alike and the error bars are quite small. Therefore, there exists a large discrepancy from the Newtonian prediction. Since the observations are repeatable and straightforward for many spiral galaxies, and Newton’s laws are very fundamental and well understood, one of the underlying assumptions of the Newtonian derivation must be wrong. The most accepted hypothesis that corrects the predicted velocity graph to match the observed graph is that most of the matter in a galaxy is actually not luminous matter, but is instead dark matter.

Which of the following correctly describes the behavior of the orbital velocity of stars in spiral galaxies as a function of radius, including the effects of dark matter?

Grows linearly in the galactic bulge and falls off exponentially outside the bulge.
Grows linearly in the galactic bulge and is constant outside the bulge.
Grows linearly in the galactic bulge and falls off like \(r^{-1/2}\) outside the bulge.
Is constant in the galactic bulge and grows linearly outside the bulge.
Falls off exponentially everywhere.

Importantly, the distribution of dark matter must not follow the same distribution as the luminous matter. The distribution of dark matter must be chosen to match the flat rotation curve at the edges of the galaxy. Dark matter is usually modeled as a halo around the galaxy that does not arrange itself into a disk, but is more similar in shape to a sphere.

The above result is one of the primary reasons physicists believe in dark matter: by adding dark matter to your galaxy, predicted rotation curves match experiment. Constructing the dark matter to have this density profile is still an area of active research, but there is little controversy as to the existence of extra, unseen matter near the edges of galaxies.

Galaxy rotation curves are not the only observational evidence for dark matter. Another effect which cannot be explained by the observed amount of luminous matter is gravitational lensing of light from certain galaxies. General relativity predicts that light bends near sources of gravitation. However, astronomical observation shows that there are regions of the universe where light from distance galaxies bends even in regions where there doesn’t appear to be any luminous matter. Other accepted evidence for dark matter includes a characteristic signature in the cosmic microwave background radiation that permeates the universe and the fact that the best models of galaxy formation in the early universe rely on the existence of a much larger amount of matter than is accounted for by the luminous matter that we have detected in the universe.

References

[1] Image from https://upload.wikimedia.org/wikipedia/commons/e/ee/VLT_image_of_the_spiral_galaxy_NGC_1187.jpg under Creative Commons licensing for reuse and modification.

[2] Image from https://en.wikipedia.org/wiki/Fraunhofer_lines#/media/File:Fraunhofer_lines.svg under Creative Commons licensing for reuse and modification.